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1. The decomposition of the higher-order homology embedding constructed from the k-Laplacian

   Chen, Yu-Chia; Meila, Marina (December 2021, Advances in neural information processing systems)

   M. Ranzato; A. Beygelzimer; Y. Dauphin; P.S. Liang; J. Wortman Vaughan (Ed.)

   The null space of the k-th order Laplacian Lk, known as the {\em k-th homology vector space}, encodes the non-trivial topology of a manifold or a network. Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data. The study of the null space embedding of the graph Laplacian L0 has spurred new research and applications, such as spectral clustering algorithms with theoretical guarantees and estimators of the Stochastic Block Model. In this work, we investigate the geometry of the k-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the {\em connected sum} of manifolds as a perturbation to the direct sum of their homology embeddings. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components. The proposed framework is applied to the {\em shortest homologous loop detection} problem, a problem known to be NP-hard in general. Our spectral loop detection algorithm scales better than existing methods and is effective on diverse data such as point clouds and images. 

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2. Integration of element specific persistent homology and machine learning for protein‐ligand binding affinity prediction

   https://doi.org/10.1002/cnm.2914  

   Cang, Zixuan; Wei, Guo‐Wei (August 2017, International Journal for Numerical Methods in Biomedical Engineering)

   Abstract Protein‐ligand binding is a fundamental biological process that is paramount to many other biological processes, such as signal transduction, metabolic pathways, enzyme construction, cell secretion, and gene expression. Accurate prediction of protein‐ligand binding affinities is vital to rational drug design and the understanding of protein‐ligand binding and binding induced function. Existing binding affinity prediction methods are inundated with geometric detail and involve excessively high dimensions, which undermines their predictive power for massive binding data. Topology provides the ultimate level of abstraction and thus incurs too much reduction in geometric information. Persistent homology embeds geometric information into topological invariants and bridges the gap between complex geometry and abstract topology. However, it oversimplifies biological information. This work introduces element specific persistent homology (ESPH) or multicomponent persistent homology to retain crucial biological information during topological simplification. The combination of ESPH and machine learning gives rise to a powerful paradigm for macromolecular analysis. Tests on 2 large data sets indicate that the proposed topology‐based machine‐learning paradigm outperforms other existing methods in protein‐ligand binding affinity predictions. ESPH reveals protein‐ligand binding mechanism that can not be attained from other conventional techniques. The present approach reveals that protein‐ligand hydrophobic interactions are extended to 40Å  away from the binding site, which has a significant ramification to drug and protein design. 

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3. Topo-Geometrically Distinct Path Computation Using Neighborhood-Augmented Graph, and Its Application to Path Planning for a Tethered Robot in 3-D

   Sahin, Alp; Bhattacharya, Subhrajit (November 2024, IEEE transactions on robotics)

   Many robotics applications benefit from being able to compute multiple geodesic paths in a given configuration space. Existing paradigm is to use topological path planning, which can compute optimal paths in distinct topological classes. However, these methods usually require nontrivial geometric constructions, which are prohibitively expensive in 3-D, and are unable to distinguish between distinct topologically equivalent geodesics that are created due to high-cost/curvature regions or prismatic obstacles in 3-D. In this article, we propose an approach to compute k geodesic paths using the concept of a novel neighborhood-augmented graph, on which graph search algorithms can compute multiple optimal paths that are topo-geometrically distinct. Our approach does not require complex geometric constructions, and the resulting paths are not restricted to distinct topological classes, making the algorithm suitable for problems where finding and distinguishing between geodesic paths are of interest. We demonstrate the application of our algorithm to planning shortest traversible paths for a tethered robot in 3-D with cable-length constraint. 

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4. A Topological Graph-Based Representation for Denoising Low Quality Binary Images

   https://doi.org/10.1109/ICCVW.2019.00222  

   Potts, Catherine; Yang, Liping; Oyen, Diane; Wohlberg, Brendt (October 2019, 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW))

   Scanned images of patent or historical documents often contain localized zigzag noise introduced by the digitizing process; yet when viewed as a whole image, global structures are apparent to humans, but not to machines. Existing denoising methods work well for natural images, but not for binary diagram images, which makes feature extraction difficult for computer vision and machine learning methods and algorithms. We propose a topological graph-based representation to tackle this denoising problem. The graph representation emphasizes the shapes and topology of diagram images, making it ideal for use in machine learning applications such as classification and matching of scientific diagram images. Our approach and algorithms provide essential structure and lay important foundation for computer vision such as scene graph-based applications, because topological relations and spatial arrangement among objects in images are captured and stored in our skeleton graph. In addition, while the parameters for almost all pixel-based methods are not adaptive, our method is robust in that it only requires one parameter and it is adaptive. Experimental comparisons with existing methods show the effectiveness of our approach. 

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5. Topological Acoustics

   https://doi.org/10.1121/AT.2021.17.3.13  

   Andrea Alu; Chiara Daraio; Pierre A. Deymier; Massimo Ruzzene (October 2021, Acoustics today)

   The field of topology studies the properties of geometric objects that are preserved under continuous deformations, for example, without cutting or gluing. A cup with a handle is topologically equivalent to a donut (or a bagel if you live in New York) because one shape can be deformed into the other while preserving their common invariant hole. Exotic topological shapes, such as vortices, knots, and mobius strips, can be globally analyzed using the mathematical tools offered by topology. The connection between topology and acoustics may appear far-fetched, yet recent developments in the field of condensed matter physics and quantum mechanics have been inspiring exciting opportunities to manipulate sound in new and unexpected ways based on topological concepts. 

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